Logistic Map

Outline Review. The period doubling bifurcation. Newton’s method and Feigenbaum’s constant Feigenbaum renormalization.

Lecture4The period doubling route in the logistic family.

Feigenbaum renormalization.

Shlomo Sternberg

Shlomo Sternberg

Lecture4 The period doubling route in the logistic family. Feigenbaum renormalization.


1 Review.

2 The period doubling bifurcation.

3 Newton’s method and Feigenbaum’s constant

4 Feigenbaum renormalization.

Shlomo Sternberg



The logistic family.

Recall that the “logistic function” is defined by

Lµ(x) := µx(1− x). (1)

Here we consider the range 0 < µ ≤ 4 so that Lµ maps the unitinterval into itself.

The fixed points of Lµ are 0 and 1− 1µ . Since L′µ(x) = µ− 2µx ,

L′µ(0) = µ, L′µ(1− 1

µ) = 2− µ.

Shlomo Sternberg



0 < µ < 1.

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For 0 < µ < 1, 0 is the only fixed point of Lµ on [0, 1] since theother fixed point, 1− 1

µ , is negative.

Shlomo Sternberg



0 is an attractive fixed point for 0 < µ < 1.

For 0 < µ < 1, 0 is the only fixed point of Lµ on [0, 1] since theother fixed point, 1− 1

µ , is negative. On this range of µ, the point

0 is an attracting fixed point since 0 < L′µ(0) = µ < 1. Underiteration, all points of [0, 1] tend to 0.

Shlomo Sternberg



µ = 1.

For µ = 1 we have

L1(x) = x(1− x) < x , ∀x > 0.

Each successive application of L1 to an x ∈ (0, 1] decreases itsvalue. The limit of the successive iterates can not be positive since0 is the only fixed point. So all points in (0, 1] tend to 0 underiteration, but ever so slowly, since L′1(0) = 1. In fact, for x < 0,the iterates drift off to more negative values and then tend to −∞.

Shlomo Sternberg



µ > 1.

For all µ > 1, the fixed point, 0, is repelling, and the unique otherfixed point, 1− 1

µ , lies in [0, 1].

Shlomo Sternberg



1 < µ < 3.

For 1 < µ < 3 we have

|L′µ(1− 1

µ)| = |2− µ| < 1,

so the non-zero fixed point is attractive, and the basin of attractionof 1− 1

µ is the entire open interval (0, 1), but the behavior isslightly different for the two domains, 1 < µ ≤ 2 and 2 < µ < 3:In the first of these ranges there is a steady approach toward thefixed point from one side or the other; in the second, the iteratesbounce back and forth from one side to the other as they convergein towards the fixed point. The graphical iteration spirals in.

Shlomo Sternberg



µ = 2 - the fixed point is superattractive.

When µ = 2, L′2(12) = 0. The fixed point, 1

2 is superattractive - theiterates zoom into the fixed point faster than any geometrical rate.

Shlomo Sternberg



µ = 3.

Much of the analysis of the preceding case applies here. Thedifferences are: the quadratic equation

−µ2x2 + (µ2 + µ)x − µ− 1.

for seeking points of period two now has a (double) root. But thisroot is 2

3 = 1− 1µ which is the fixed point. So there is still no point

of period two other than the fixed points. The iterates continue tospiral in, but now ever so slowly since L′µ(2

3) = −1.

Shlomo Sternberg



µ > 3, points of period two appear.

For µ > 3 we have

L′µ(1− 1

µ) = 2− µ < −1

so both fixed points, 0 and 1− 1µ are repelling. But now

−µ2x2 + (µ2 + µ)x − µ− 1. has two real roots which are

p2± =1

2+

1

2µ± 1

2µ

√(µ+ 1)(µ− 3).

Shlomo Sternberg



µ = 3.3, graphs of y = x , y = Lµ(x), y = L2µ(x).

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Notice that now the graph of L2µ has four points of intersection

with the line y = x : the two (repelling) fixed points of Lµ and twopoints of period two.

Shlomo Sternberg



The derivative of L2µ at these points of period two is given by

(L2µ)′(p2±) = L′µ(p2+)L′µ(p2−)

= (µ− 2µp2+)(µ− 2µp2−)

= µ2 − 2µ2(p2+ + p2−) + 4µ2p2+p2−

= µ2 − 2µ2(1 +1

µ) + 4µ2 × 1

µ2(µ+ 1)

= −µ2 + 2µ+ 4.

This last expression equals 1 when µ = 3 as we already know. Itdecreases as µ increases reaching the value −1 when µ = 1 +

√6.

Shlomo Sternberg



Graphical iteration for µ = 3.3, nine steps.

Notice the “spiraling out” from the fixed point.Shlomo Sternberg



Graphical iteration for µ = 3.3, twenty five steps.

The attractive period two points become evident.Shlomo Sternberg



3 < µ < 1 +√

6.

In this range the fixed points are repelling and both period twopoints are attracting. There will be points whose images end up,after a finite number of iterations, on the non-zero fixed point. Allother points in (0, 1) are attracted to the period two cycle. Weomit the proof.

Shlomo Sternberg



Superattracting period two points.

Notice also that there is a unique value of µ in this range where

p2+(µ) =1

2.

Indeed, looking at the formula for p2+ we see that this amounts tothe condition that

√(µ+ 1)(µ− 3) = 1 or

µ2 − 2µ− 4 = 0.

The positive solution to this equation is given by µ = s2 where

s2 = 1 +√

5.

Shlomo Sternberg



Ats2 = 1 +

√5

the period two points are superattracting, since one of themcoincides with 1

2 which is the maximum of Ls2 .

Shlomo Sternberg



3.449499... < µ < 3.569946....

Once µ passes 1 +√

6 = 3.449499... the points of period twobecome unstable and (stable) points of period four appear. Initiallythese are stable, but as µ increases they become unstable (at thevalue µ = 3.544090...) and bifurcate into period eight points,initially stable.

Shlomo Sternberg



Graphical iteration for µ = 3.46, twenty five steps.

The attractive period four points become evident.Shlomo Sternberg



Reprise.

The total scenario so far, as µ increases from 0 to about 3.55, is asfollows: For µ < b1 := 1, there is no non-zero fixed point. Past thefirst bifurcation point, b1 = 1, the non-zero fixed point hasappeared close to zero. When µ reaches the first superattractivevalue , s1 := 2, the fixed point is at .5 and is superattractive. As µincreases, the fixed point continues to move to the right. Just afterthe second bifurcation point, b2 := 3, the fixed point has becomeunstable and two stable points of period two appear, one to theright and one to the left of .5.

Shlomo Sternberg



Just after the second bifurcation point, b2 := 3, the fixed point hasbecome unstable and two stable points of period two appear, oneto the right and one to the left of .5.

The leftmost period two point moves to the right as we increase µ,and at µ = s2 := 1 +

√5 = 3.23606797... the point .5 is a period

two point, and so the period two points are superattractive. Whenµ passes the second bifurcation value b2 = 1 +

√6 = 3.449.. the

period two points have become repelling and attracting period fourpoints appear.

Shlomo Sternberg



In fact, this scenario continues. The period 2n−1 points appear atbifurcation values bn. They are initially attracting, and becomesuperattracting at sn > bn and become unstable past the nextbifurcation value bn+1 > sn when the period 2n points appear. The(numerically computed) bifurcation points and superstable pointsare tabulated as:

Shlomo Sternberg



n bn sn1 1.000000 2.0000002 3.000000 3.2360683 3.449499 3.4985624 3.544090 3.5546415 3.564407 3.5666676 3.568759 3.5692447 3.569692 3.5697938 3.569891 3.5699139 3.569934 3.569946∞ 3.569946 3.569946

Shlomo Sternberg



The values of the bn are obtained by numerical experiment. Later,we shall describe a method for computing the sn using Newton’smethod.

Shlomo Sternberg



The graph of the first four bifurcations.

Shlomo Sternberg



I should explain how this figure was drawn: For each value of rranging in steps of .005 from 0 to 3.55 the values of L◦kr (x0) werecomputed for 100 values of k (where x0 was chosen as 0.4). Thenonly the last 30 values were kept, and these were plotted against r.

In the next slide I will give the MATLAB program for doing this,modified very slightly from Lynch.

Shlomo Sternberg



clear, itermax=100;finalits=30;finits=itermax-(finalits-1);for r=0:0.005:4x=0.4; xo=x; for n=2:itermaxxn=r*xo*(1-xo);x=[x xn];xo=xn;endplot(r*ones(finalits),x(finits:itermax),’.’,’MarkerSize’,1)hold onendfsize=15; set(gca,’xtick’,[0:1:4],’FontSize’,fsize),set(gca,’ytick’,[0,0.5,1],’FontSize’,fsize)xlabel(’mu’,’FontSize’,fsize), ylabel(’itx’,’FontSize’,fsize), hold off

Shlomo Sternberg



We should point out that this is still just the beginning of thestory. For example, an attractive period three cycle appears atabout 3.83. We shall come back to all of these points, but first goback and discuss theoretical problems associated to bifurcations, inparticular, the period doubling bifurcation.

Shlomo Sternberg



Review.

In the last lecture we studied the general theory of the perioddoubling bifurcation, where an attractive fixed point becomesrepelling and two attractive double points appear, as illustrated inthe following diagram:

Shlomo Sternberg



-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-1

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! attracting fixed pointrepelling fixed point!

" attracting double point

µ

Shlomo Sternberg



The first period doubling bifurcation in the logistic family

To visualize the phenomenon in the logistic family, we plotted thefunction L◦2µ for the values µ = 2.9 and µ = 3.3. For µ = 2.9 thecurve crosses the diagonal at a single point, which is in fact a fixedpoint of Lµ and hence of L◦2µ . This fixed point is stable. Forµ = 3.3 there are three crossings. The non-zero fixed point of Lµhas derivative smaller than −1, and hence the corresponding fixedpoint of L◦2µ has derivative greater than one. The two othercrossings correspond to the stable period two orbit.

Shlomo Sternberg



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Shlomo Sternberg



When applied to F ◦2µ each of the two branches (the two fixedpoints of F ◦2µ ) splits into two attractive points of period four as inour diagram. Notice that the two branches split at the same valueof µ since this is the point where F ◦2µ has derivative -1, and thederivative of F ◦2µ is the same at each of the two fixed points of F ◦2µ .

Shlomo Sternberg



Also, in the last lecture, we proved a general theorem givingconditions that a period doubling bifurcation occur. I will notrepeat this here.

Shlomo Sternberg



Although the bifurcation values bn for the logistic family are hardto compute except by numerical experiment, the superattractivevalues can be found by applying Newton’s method to find thesolution, sn, of the equation

L◦2n−1

µ (1

2) =

1

2, Lµ(x) = µx(1− x). (2)

This is the equation for µ which says that 12 is a point of period

2n−1 of Lµ. Of course we want to look for solutions for which 12

does not have lower period.

Shlomo Sternberg



So we set

P(µ) = L2n−1

µ (1

2)− 1

2

and apply the Newton algorithm

µk+1 = N (µk), N (µ) = µ− P(µ)

P ′(µ).

with ′ now denoting differentiation with respect to µ.

Shlomo Sternberg



As a first step, must compute P and P ′. For this we define thefunctions xk(µ) recursively by

x0 ≡1

2, x1(µ) = µ

1

2(1− 1

2), xk+1 = Lµ(xk),

so, we have

x ′k+1 = [µxk(1− xk))]′

= xk(1− xk) + µx ′k(1− xk)− µxkx ′k

= xk(1− xk) + µ(1− 2xk)x ′k .

Shlomo Sternberg



LetN = 2n−1

so that

P(µ) = xN −1

2, P ′(µ) = x ′N(µ).

Shlomo Sternberg



Thus, at each stage of the iteration in Newton’s method wecompute P(µ) and P ′(µ) by running the iteration scheme

xk+1 = µxk(1− xk) x0 = 12

x ′k+1 = xk(1− xk) + µ(1− 2xk)x ′x x ′0 = 0

for k = 0, . . . ,N − 1. We substitute this into Newton’s method,get the next value of µ, run the iteration to get the next value ofP(µ) and P ′(µ) etc.

Shlomo Sternberg



Suppose we have found s1, s2, ...., sn. What should we take as theinitial value of µ? Define the numbers δn, n ≥ 2 recursively byδ2 = 4 and

δn =sn−1 − sn−2

sn − sn−1, n ≥ 3. (3)

We have already computed

s1 = 2, s2 = 1 +√

5 = 3.23606797 . . . .

We take as our initial value in Newton’s method for finding sn+1

the value

µn+1 = sn +sn − sn−1

δn.

The following facts are observed:

Shlomo Sternberg



For each n = 3, 4, . . . , 15, Newton’s method converges very rapidly,with no changes in the first nineteen digits after six applications ofNewton’s method for finding s3, after only one application ofNewton’s method for s4 and s5, and at most four applications ofNewton’s method for the computation of each of the remainingvalues.

Suppose we stop our calculations for each sn when there is nofurther change in the first 19 digits, and take the computed valuesas our sn. These values are strictly increasing. In particular thisimplies that the sn we have computed do not yield 1

2 as a point oflower period.

Shlomo Sternberg



The sn approach a limiting value, 3.569945671205296863.

The δn approach a limiting value,

δ = 4.6692016148.

This value is known as Feigenbaum’s constant. While the limitingvalue of the sn is particular to the logistic family, δ is “universal” inthe sense that it applies to a whole class of one dimensionaliteration families. We shall go into this point in the next section,where we will see that this is a “renormalization group”phenomenon.

Shlomo Sternberg



We have already remarked that the rate of convergence to thelimiting value of the superstable points in the period doublingbifurcation, Feigenbaum’s constant, is universal, i.e. not restrictedto the logistic family. That is, if we let

δ = 4.6692....

denote Feigenbaum’s constant, then the superstable values sr inthe period doubling scenario satisfy

sr = s∞ − Bδ−r + o(δ−r )

where s∞ and B depend on the specifics of the family, but δapplies to a large class of such families.

Shlomo Sternberg



There is another “universal” parameter in the story. Suppose thatour family fµ consists of maps with a single maximum, Xm, so thatXm must be one of the points on any superstable periodic orbit.(In the case of the logistic family Xm = 1

2 .) Let dr denote thedifference between Xm an the next nearest point on the superstable2r orbit; more precisely, define

dr = f 2r−1

sr (Xm)− Xm.

Then dr ∼ D(−α)r where

α.

= 2.5029...

is again universal. This would appear to be a scale parameter (inx) associated with the period doubling scenario.

Shlomo Sternberg



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To understand this scale parameter, examine the central portion ofthe figure, and observe that the graph of L◦2µ looks like an(inverted and) rescaled version of Lµ, especially if we allow achange in the parameter µ.

Shlomo Sternberg



Before going into the rescaling operator on functions, I would liketo give an elementary formulation of what is going on, following abeautiful paper by S.N. Coppersmith A simpler derivation ofFeigenbaums renormalization group equation for theperiod-doubling bifurcation sequence which appeared in theAmerican Journal of Physics, Vol 67 (1999) 53. Also see herpowerpoint presentation available on the web.

Shlomo Sternberg



Take µ = 3.569946 and plot the values Ljµ(.5)− .5 against j . Then

plot every other value with the ordinate upside down and rescaledby a factor of 2.502 907 9. The graphs look the same:

every j plotted. every other j plotted ordinateupside down.

Shlomo Sternberg



We can check this numerically by computing the vector y (say oflength 21) with y(1) = .5 and y(n + 1) = Lµ(y(n)), then thevector z with z(i) = y(i)− .5 and then comparing the first 11entries of z with the vector k obtained by taking every other entryof z and multiplying by -2.502. The results are:

Shlomo Sternberg



0 00.3925 0.3939−0.1574 −0.1566

0.3040 0.29760.0626 0.06260.3785 0.3783−0.1190 −0.1189

0.3420 0.3380−0.0250 −0.0250

0.3903 0.3914−0.1512 −0.1505

Shlomo Sternberg



The existence of the scaling (together with some argumentation)determines the scale parameter as follows: We presume to have

−αz2j = zj

which, replacing j by j + 1 gives −αz2j+2 = zj+1. Writezj+1 = g(zj). The second equation gives −αg(g(z2j) = g(zj) andwe can substitute z2j = −zj/α from the first equation to get

−αg(g(−zj/α)) = g(zj).

If we expect this to hold not just for zj but for all values of z weget the functional equation:

Shlomo Sternberg



−αg(g(−z/α)) = g(z).

If we assume that g has a power series expansion, and we computeup to terms of second order in z , we get an approximate value forα.

Shlomo Sternberg



The rescaling is centered at the maximum, so in order to avoidnotational complexity, let us shift this maximum (for the logisticfamily) to the origin by replacing x by y = x − 1

2 . In the newcoordinates the logistic map is given by

y 7→ Lµ(y +1

2)− 1

2= µ(

1

4− y2)− 1

2.

Let R denote the operator on functions given by

R(h)(y) := −αh(h(y/(−α))). (4)

In other words, R sends a map h into its iterate h ◦ h followed by arescaling.

Shlomo Sternberg



We are going to not only apply the operator R, but also shift theparameter µ in the maps

hµ(y) = µ(1

2− y2)− 1

2

from one supercritical value to the next. So for eachk = 0, 1, 2, . . . we set

gk0 := hsk

and then define

gk,1 = Rgk+1,0

gk,2 = Rgk+1,1

gk,3 = Rgk+2,1

......

Shlomo Sternberg



It is observed (numerically) that for each k the functions gk,r

appear to be approaching a limit, gk i.e.

gk,r → gk .

Sogk(y) = lim(−α)rg2r

sk+r(y/(−α)r ).

Hence

Rgk = lim(−α)r+12r+1gsk+r(y/(−α)r+1) = gk−1.

Shlomo Sternberg



It is also observed that these limit functions gk themselves areapproaching a limit:

gk → g .

Since Rgk = gk−1 we conclude that

Rg = g ,

i.e. g is a fixed point for the Feigenbaum renormalization operatorR.

Shlomo Sternberg



Notice that rescaling commutes with R: If S denotes the operator(Sf )(y) = cf (y/c) then

R(Sf )(y) = −α(c(f (cf (y/(cα))/c) = S(R)f (y).

So if g is a fixed point, so is Sg . We may thus fix the scale in g byrequiring that

g(0) = 1.

Shlomo Sternberg



The hope was then that there would be a unique function g(within an appropriate class of functions) satisfying

Rg = g , g(0) = 1,

or, spelling this out,

g(y) = −αg◦2(−y/α), g(0) = 1. (5)

Notice that if we knew the function g , then setting y = 0 in (5)gives

1 = −αg(1)

orα = −1/g(1).

Shlomo Sternberg



In other words, assuming that we were able to establish all thesefacts and also knew the function g , then the universal rescalingfactor α would be determined by g itself. Feigenbaum assumedthat g has a power series expansion in x2 took the first seventerms in this expansion and substituted in (5). He obtained acollection of algebraic equations which he solved and then derivedα close to the observed “experimental” value. Indeed, if wetruncate (5) we will get a collection of algebraic equations. Butthese equations are not recursive, so that at each stage oftruncation modification is made in all the coefficients, and also thenature of the solutions of these equations is not transparent.

Shlomo Sternberg



So theoretically, if we could establish the existence of a uniquesolution to (5) within a given class of functions the value of α isdetermined. But the numerical evaluation of α is achieved by therenormalization property itself, rather than from g(1) which is notknown explicitly.The other universal constant associated with the period doublingscenario, the constant δ was also conjectured by Feigenbaum to beassociated to the fixed point g of the renormalization operator;this time with the linearized map J, i.e. the derivative of therenormalization operator at its fixed point.

Shlomo Sternberg



Later on we will see that in finite dimensions, if the derivative J ofa non-linear transformation R at a fixed point has k eigenvalues> 1 in absolute value, and the rest < 1 in absolute value, thenthere exists a k-dimensional R invariant surface tangent at thefixed point to the subspace corresponding to the k eigenvalueswhose absolute value is > 1. On this invariant manifold, the mapR is expanding. Feigenbaum conjectured that for the operator R(acting on the appropriate infinite dimensional space of functions)there is a one dimensional “expanding” submanifold, and that δ isthe single eigenvalue of J with absolute value greater than 1.

Shlomo Sternberg



In the course of the past thirty five years, these conjectures ofFeigenbaum have been verified using high powered techniques fromcomplex analysis, thanks to the combined effort of suchmathematicians as Douady, Hubbard, Sullivan, McMullen, and. . . .

Shlomo Sternberg


Logistic Map

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